Answer:
[tex](f.g)(x) = 3x^3-3x^2-24x+36[/tex]
Domain of (f.g)(x) = All real numbers
[tex](\frac{f}{g})(x) = \frac{x+3}{3}[/tex]
Domain of (f/g)(x) = All Real Numbers
Step-by-step explanation:
Given functions are:
[tex]f(x) = x^2+x-6\\g(x) = 3x-6[/tex]
We have to calculate f.g and f/g
In order to find f.g we have to multiply both functions
[tex](f.g)(x) = (x^2+x-6)(3x-6)\\= 3x(x^2+x-6) - 6(x^2+x-6)\\= 3x^3+3x^2-18x-6x^2-6x+36\\= 3x^3+3x^2-6x^2-18x-6x+36\\=3x^3-3x^2-24x+36[/tex]
The domain of (f.g)(x) is all real numbers
Now
[tex](\frac{f}{g})(x) = \frac{f(x)}{g(x)}\\= \frac{x^2+x-6}{3x-6}\\=\frac{x^2+3x-2x-6}{3x-6}\\=\frac{x(x+3)-2(x+3)}{3(x-2)}\\=\frac{(x+3)(x-2)}{3(x-2)}\\=\frac{x+3}{3}[/tex]
The domain of (f/g)(x) is all real numbers.
Hence,
[tex](f.g)(x) = 3x^3-3x^2-24x+36[/tex]
Domain of (f.g)(x) = All real numbers
[tex](\frac{f}{g})(x) = \frac{x+3}{3}[/tex]
Domain of (f/g)(x) = All Real Numbers
